54 PROBLEMS IN THE CONDUCTION OF HEAT
If in (12) we put q^o-da and integrate with respect to a from 0 t we obtain a solution which must coincide with (7) when in the luttt substitute z for x. Thus
o a particular case of one of Weber's integrals*.
It may be worth while to consider briefly the problem of initin stantaneous sources distributed over the plane (Ģ=0) in a more ge manner. In rectangular coordinates the typical distribution is such tha rate per unit of area is
a- cos It; . cos mij ............................... (]
If Ave assume that at x, y, z and time t, v is proportional to cos lx . eo the general differential equation (1) gives
_ __,
*
or ~
so that, as for conduction in one dimension,
0-Z2/.K
v = A cos lx cos my e-P+m*>t j , .................. (*
Y t
r+cc
and vdz = 2^/7r.Acoslxcos my e
J 00
Putting t=Q, and comparing with (14), we see that
cr
2V7T'
By means of (2) the solution at time t may be built up from (14) this way, by aid of the well-known integral
e~a*x- cos 2crc <Ģe = e-flS/Ŧ2 t
I -x a ' ....................^
we may obtain (15) independently.
The process is of more interest in its application to polar coordii If we suppose that v is proportional to cos nQ . Jn (kr),
d?v I dv 1 d-v also*(?l)aybelonwd ******' ^^ ^^ P' ^ *^ (16°^ Put "=°' ^d integrate with respect to z from oo to + oo , we may recover (9).(9) are of the standard form if we take